Question: The lifespans of seals in a particular zoo are normally distributed. The average seal lives $14.9$ years; the standard deviation is $3.4$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a seal living between $4.7$ and $11.5$ years.
Solution: $14.9$ $11.5$ $18.3$ $8.1$ $21.7$ $4.7$ $25.1$ $99.7\%$ $68\%$ $15.85\%$ $15.85\%$ We know the lifespans are normally distributed with an average lifespan of $14.9$ years. We know the standard deviation is $3.4$ years, so one standard deviation below the mean is $11.5$ years and one standard deviation above the mean is $18.3$ years. Two standard deviations below the mean is $8.1$ years and two standard deviations above the mean is $21.7$ years. Three standard deviations below the mean is $4.7$ years and three standard deviations above the mean is $25.1$ years. We are interested in the probability of a seal living between $4.7$ and $11.5$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the seals will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $68\%$ of the seals will have lifespans within 1 standard deviation of the mean. The probability of a particular seal living between $4.7$ and $11.5$ years is $\color{orange}{15.85\%}$.